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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Exam 3 -- December 5, 2012}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible 
regular points and 10 Extra Credit points.
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\noindent
I.  In an RSA public key cryptosystem, the public key information
is $m = 323$ and $e = 13$.  Messages consisting of capital roman
letters and blanks are encoded as 3-digit blocks $000, 001, \cdots, 026$
(with blank = $000$, $A = 001$, $B = 002$, ... , $Z = 026$)
and encrypted as 3-digit blocks.
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\item{A)}  (15) How would the plaintext symbol $N$ be encrypted?
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\item{B)}  (15) What is the (secret) decryption exponent $d$?
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\noindent
II. (20) Let
$$H = \left\{\pmatrix{a & b \cr 0 & 1\cr} \mid a,b \in {\bf R}\ {\rm and}\ a \ne 0\right\}$$
Is $H$ a group under the operation of matrix multiplication?
If so, give a proof.  If not, say which of the group properties fail.
{\it Note:  We discussed a general rule
for finding the multiplicative inverse of a $2\times 2$ matrix on a problem set
earlier in the semester.  If you don't
remember it, you can ``buy'' it from me during the exam in return 
for a reduction of 5 points on your score for this problem.}
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\noindent
III. 
\item{A)} (15) Let $G$ be a cyclic group with generator $a$.  Show
that every subgroup of $G$ is also cyclic.
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\item{} The next parts of this question refer to ${\bf Z}_{24}$, which 
is a cyclic group under addition mod 24.
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\item{B)} (10) How many different subgroups does ${\bf Z}_{24}$ contain,
including ${\bf Z}_{24}$ itself and $\{[0]\}$?
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\item{C)} (15) Show that if $\gcd(a,24) = 1$, then $\phi : {\bf Z}_{24} \to {\bf Z}_{24}$
defined by $\phi([x]) = [ax]$ is a 1-1 and onto group homomorphism. 
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\noindent
IV.  (10) Let $G$ be a group with operation written as multiplication, 
let $H$ be a subgroup of $G$, and let $a \in G$ be a fixed element.
Let $aH = \{ah \mid h \in H\}$.  Show that $aH$ is a subgroup of $G$ if and only 
if $a \in H$.         
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\noindent
{\it Extra Credit} (10)  A group $G$ is generated by elements $x,y$ satisfying 
$x^n = e$, $y^2 = e$, and $y x = x^{n-1} y$.  Show that all of the elements $x^\ell y$ for
$\ell = 0,1,\ldots,n-1$ have order 2.  

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{\it The General Pattern for $2\times 2$ Matrix Inverses}
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\noindent
If 
$$A = \pmatrix{a & b \cr c & d\cr}$$
then $A$ has a inverse matrix (for multiplication) if and only 
if $\det(A) = ad - bc \ne 0$, and then
$$A^{-1} = \pmatrix{ {d\over ad - bc} & {-b\over ad - bc}\cr
                     {-c\over ad - bc} & {a\over ad - bc}\cr}$$
\bye




